A fundamental control problem is determining a suitable input signal to drive a physical component from a first position (at time ti) to a second position (at time tf) during an output-transition time interval, ti−tf. This problem arises in controlling a driver for almost any form of physical component that is rapidly positioned. In many such control applications, the controlled component must be precisely positioned both at a first position, before the output-transition time interval, and at a second position, after the output-transition time interval. Performing fast maneuvers with flexible structures can introduce severe vibration problems, resulting in loss of positioning precision after the completion of an output transition and inefficiency in the positioning process. For example, when positioning a read/write head of a disk drive, read or write operations cannot be performed before or after an output transition that moves the head, if the output position is not precisely maintained at the desired track. Residual vibrations arising due to the elastic flexibility of the head support structure may require a prohibitively long time to settle out, increasing the working cycle time, since the read/write head cannot perform a next task until the vibrations are reduced to an acceptable level. Accordingly, it is very important in such control applications to achieve output transitions without residual vibrations.
As opposed to the problem of changing the output-point on a flexible structure, the problem of changing the complete configuration, i.e., the state of system such as a flexible structure, has been well studied in prior literature. These techniques, which solve the state-to-state transition problem (sometimes referred to as the state-transition problem), can also be used to find a solution to the output-transition problem. In particular, output transitions without residual vibrations can be obtained by requiring that the flexible system maneuver between equilibrium configurations (rigid-body rest configurations). These rest states are chosen to achieve initial and final output values at the beginning and end of the output transition. Once the boundary states at the beginning and end of output transition are chosen to be the rest states, then a solution to the output-transition problem can be found by solving the standard, optimal (e.g., minimum-energy), state-transition problem from the initial state to the final state, which is referred to as the “rest-to-rest state-transition approach.” However, the solution found with this choice of the boundary states may not lead to an optimal output-transition. On the other hand, arbitrary choices of the boundary states are also not acceptable, because they may not allow the output to be maintained at a constant value after the completion of the output-transition, i.e., without residual vibrations, for any choice of the input. Therefore, standard optimal state-to-state transition approaches cannot be used to directly solve the optimal output-transition problem.
Clearly, a better approach is needed that permits the input required to achieve an optimal output transition. In addition to enabling a minimum input energy to be achieved, it would also be desirable to enable other optimal criteria, such as achieving a minimum time for a transition, to be applied in determining the input needed to achieve the desired goal.